Spectral properties of operators of the theory of harmonic potential (Q1358445)

Let \(\Omega\) be a simply connected finite domain in \({\mathbb{R}}^3\) bounded by a Lyapunov surface \(S\). The authors study spectral properties of the following operators of the classical potential theory, on \(L^2(S)\): \[ (B\varphi)(x):=\int_S\varphi(y){\partial|x-y|^{-1}\over \partial n_x}dS_y \] and its adjoint \(B^\ast\). In particular, they show that there exists an ellipsoid of rotation such that zero belongs to the point spectrum of both operators \(B\) and \(B^\ast\). The authors also prove that for any \(b\in [0,2\pi)\) there exists a smooth surface such that \(b\) is an eigenvalue for both operators \(B\) and \(B^\ast\) defined on \(S\).